Effective mass of electron in monolayer graphene : Electron-phonon interaction

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Effective mass of electron in monolayer

graphene: Electron-phonon interaction

E Tiras, S Ardali, T Tiras, E Arslan, S Cakmakyapan, O Kazar, Jawad Hassan, Erik Janzén

and E Ozbay

Linköping University Post Print

N.B.: When citing this work, cite the original article.

Original Publication:

E Tiras, S Ardali, T Tiras, E Arslan, S Cakmakyapan, O Kazar, Jawad Hassan, Erik Janzén

and E Ozbay, Effective mass of electron in monolayer graphene: Electron-phonon interaction,

2013, Journal of Applied Physics, (113), 4.

http://dx.doi.org/10.1063/1.4789385

Copyright: American Institute of Physics (AIP)

http://www.aip.org/

Postprint available at: Linköping University Electronic Press

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Effective mass of electron in monolayer graphene: Electron-phonon

interaction

E. Tiras, S. Ardali, T. Tiras, E. Arslan, S. Cakmakyapan et al.

Citation: J. Appl. Phys. 113, 043708 (2013); doi: 10.1063/1.4789385 View online: http://dx.doi.org/10.1063/1.4789385

View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v113/i4 Published by the American Institute of Physics.

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Effective mass of electron in monolayer graphene: Electron-phonon

interaction

E. Tiras,1,a)S. Ardali,1T. Tiras,1E. Arslan,2S. Cakmakyapan,2O. Kazar,2J. Hassan,3 E. Janzen,3and E. Ozbay2

1

Department of Physics, Faculty of Science, Anadolu University, Yunus Emre Campus, 26470 Eskisehir, Turkey

2

Nanotechnology Research Center, Departments of Physics and Electrical and Electronics Engineering, Bilkent University, Bilkent, 06800 Ankara, Turkey

3

Departments of Physics, Chemistry and Biology, Linkoping University of Technology, S-581 83 Linkoping, Sweden

(Received 4 September 2012; accepted 8 January 2013; published online 25 January 2013)

Shubnikov-de Haas (SdH) and Hall effect measurements performed in a temperature range between 1.8 and 275 K, at an electric field up to 35 kV m1and magnetic fields up to 11 T, have been used to investigate the electronic transport properties of monolayer graphene on SiC substrate. The number of layers was determined by the use of the Raman spectroscopy. The carrier density and in-plane effective mass of electrons have been obtained from the periods and temperature dependencies of the amplitude of the SdH oscillations, respectively. The effective mass is in good agreement with the current results in the literature. The two-dimensional (2D) electron energy relaxations in monolayer graphene were also investigated experimentally. The electron temperature (Te) of hot

electrons was obtained from the lattice temperature (TL) and the applied electric field dependencies

of the amplitude of SdH oscillations. The experimental results for the electron temperature dependence of power loss indicate that the energy relaxation of electrons is due to acoustic phonon emission via mixed unscreened piezoelectric interaction and deformation-potential scattering.

VC _{2013 American Institute of Physics. [}_{http://dx.doi.org/10.1063/1.4789385}_]

I. INTRODUCTION

The discovery of graphene, a dense honeycomb two-dimensional structure formed by a monolayer or a few layers of carbon atoms, has led to the intense research activ-ities aimed at understanding the electronic properties of this truly two-dimensional (2D) electronic system.1It is a mass-less and gapmass-less Dirac quasi-particle system with roughly linear energy dispersion. Its unique physical properties make it an ideal system to look for new physics, and it is a promising candidate for applications from gas sensors to touch screens.

The Shubnikov-de Haas (SdH) oscillations in magneto-resistance have been extensively in the investigations of the electronic transport properties of 2D structures at low tem-perature. The SdH effect has been proven to be a powerful tool for determining the in-plane effective mass (m*) and electron-phonon interactions.2–5Recently, the electron effec-tive mass in monolayer graphene has been obtained from the temperature dependence of the amplitude of SdH oscilla-tions.6–8 They reported the electron effective mass of graphene samples depending on the square root of the 2D carrier concentration.

The determination of the temperature of electrons, under electric-field heating conditions in the steady state, provides useful information about the electron-phonon interactions involved in the energy relaxation process. Electron energy loss rates in graphene samples have been investigated

theoretically9–12 and experimentally.8,13–18 Most previous experimental works on energy loss rates in graphene have been conducted using optical excitation at much higher ener-gies13–15with non-equilibrium carrier distributions and car-rier temperatures in excess of 5000 K where the carcar-rier lifetime has saturated. In addition, the electron energy loss rates in graphene samples have been investigated using elec-trical measurements performed in a high-current regime at temperatures between 4 and 250 K.16,17Recently, Tanet al.8 and Baker et al.18 investigated the carrier temperature dependence of the energy loss rate in graphene by using the amplitude of Shubnikov de Haas oscillations as a func-tion of the electric field. It was reported that the carrier energy loss in graphene found to be proportional to fourth order of the electron temperature at carrier temperatures up to 100 K, indicating that the energy relaxation of electrons is due to acoustic phonon emission via deformation potential coupling.18,19

An in-depth understanding of the fundamental optical and electronic properties is yet to be established for the design and development of functional devices. The objec-tive of this work is to determine the in-plane effecobjec-tive mass of electrons in monolayer graphene. This is achieved by systematically measuring the temperature dependence of Shubnikov de Haas (SdH) oscillations in magnetoresistance for monolayer graphene. Furthermore, the temperature of the hot electrons (Te) of the monolayer graphene sample

and the corresponding power loss (P) have been determined as a function of the applied electric field using the SdH effect method.

a)_{Electronic mail: etiras@anadolu.edu.tr.}

0021-8979/2013/113(4)/043708/7/$30.00 113, 043708-1 VC2013 American Institute of Physics

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II. EXPERIMENTAL PROCEDURE

Experiments were carried out on the epitaxial graphene grown on (10 10) mm2nominally on-axis 4H-SiC (0001), Si-face chemo-mechanically polished substrates. All the sub-strates were obtained from a single 4-in. wafer to avoid the influence of large variations of unintentional off-cut from wafer to wafer. Three samples were grown during each growth run to have similar graphene. This was mainly to facilitate device processing and to understand the influence of different processing steps on the electrical properties of gra-phene. The optimized surface preparation and growth process were used to obtain 1-2 layers of graphene.In situ etching, graphene growth, and hydrogen intercalation were performed on samples during the same and single growth sequence with-out exposing samples to air. A uniquein situ surface prepara-tion method was adopted from the on-axis homoepitaxial growth of 4H-SiC. The samples were exposed to a mixture of silane and hydrogen (0.006% silane in hydrogen) for 10 min at 1400C. The following graphene growth was performed in vacuum (5–9 106mbar) at 1400C for 1 h. After graphene growth, the samples were cooled to below 500C in vacuum. The growth cell was then filled with hydrogen to a pressure of 500 mbar, and the intercalation process was made at 700C for 1 h. For the transport measurements by using the Hall bar geometry, we have designed and fabricated a photomask with electron beam lithography in order to perform each fab-rication step with optical lithography. Ohmic contacts were fabricated with the reverse lithography technique. After the development, 20 nm titanium and 100 nm gold were deposited by an electron beam evaporator, followed by the standard lift-off process. The mesa lithography step was performed in order to preserve the active graphene region, while etching the rest of the graphene on the surface with O2plasma. After

the etching process, the 500 lm by 1100 lm active graphene region was obtained. Interconnect metal lithography was per-formed by using a 30 nm/220 nm Ti/Au metal pair. Finally, devices were bonded for Hall measurements.

The longitudinal resistance (Rxx) along the applied

cur-rent measurements were carried out as functions of (i) the applied electric fieldF at a fixed lattice temperature TL0 and

(ii) the lattice temperatureTLat a fixed electric fieldF0that

was low enough to ensure ohmic conditions and hence to avoid carrier heating. In the experiments, a conventional dc technique in combination with a constant current source (Keithley 2400) and a nanovoltmeter (Keithley 2182 A) in a cryogen free superconducting magnet system (Cryogenics Ltd., Model No. J2414) were used. The current (I) flow was in the plane of the electron gas. Steady magnetic fields up to 11 T were applied perpendicular to the plane of the samples and, therefore, to the plane of 2D electron gas. All the meas-urements were taken in the dark. TheRxxand the Hall

resist-ance (Rxy) were measured as a function of temperature from

1.8 to 275 K. A static magnetic field (B¼ 1 T) was applied to the sample perpendicular to the current plane. The Hall mobility (lH) and the sheet carrier density (NH) were

obtained using the following equations: Rxy¼ B NHe ; (1) l_H¼ L bNHeRxx ¼est m; (2)

where st is the transport lifetime, b (¼0.6 mm) and L

(¼1 mm) are the width and length of the Hall bar, respec-tively. The applied electric field was also obtained using the longitudinal resistance measured atB¼ 0 T (RxxðB ¼ 0Þ

¼ 6113 X) in the following equation: F¼RxxðB ¼ 0Þ

L : (3)

In the applied electric field dependent magnetoresistance measurements, applied current was applied along the length of the sample in the range ofI¼ 1 lA to 8 mA.

The Raman spectra were obtained at room temperature by using the Bruker Optics FT-Raman Scope III system. As an excitation source, a wavelength of 633 nm (1.96 eV) was applied in the sample growth direction.

III. RESULTS AND DISCUSSION

Figure1shows the room-temperature Raman spectra for the graphene recorded in the grown-axis backscattering con-figurations. For the graphene layer on SiC, it is apparently dif-ficult to distinguish the G and D peaks in the spectra because their intensities are much smaller than those of the surround-ing peaks from the SiC substrate.20The sharp and strong peak at 775.0 cm1 is the E2planar optical, 962.5 cm1 is an A1

longitudinal optical (LO) mode, and 201.5 cm1which is an E2planar or transverse acoustic mode of SiC.21 The

defect-induced D band peak, which should be near 1350 cm1, is absent for these layers, attesting to the high crystalline quality of graphene layers.20 The small G peak is also located in between the SiC peaks at 1500 and 1700 cm1.20Only the 2D peaks are well isolated around 2600-2750 cm1and are out of the range of the SiC peaks.22 It has been shown that the Raman scattering fingerprint of graphene mono and multi layers lies in the 2D feature, whose shape strongly depends on the interlayer coupling. Graphene shows a single component 2D band while a multi-components 2D band is observed upon increasing the number of layers in graphitic structures. In our sample, the 2D peak at 2668.5 cm1indicates that the sample is a monolayer of graphene.

FIG. 1. Raman spectra in monolayer graphene on SiC substrate. The enlarged 2D-band region is also shown in the inserted figure.

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The temperature dependence of the sheet carrier density and Hall mobility in the graphene is plotted in Fig.2. At low temperatures, the sheet carrier density remains practically constant up to a temperature of 100 K. At higher tempera-tures, the sheet carrier density increases monotonically with increasing temperature possibly due to thermally generated carriers located outside the channel. A similar behavior of carrier density with temperature, although less pronounced, was reported previously for 2D structures such as modulation-doped GaAs/Ga_1xAlxAs heterojunctions.23–25

The Hall mobility of electrons in the graphene sample increases monotonically with a decreasing temperature from room temperature, begins to level off at about 100 K, and sat-urates at about 50 K (see Fig.1). This behavior reflects the 2D character of the electrons in the channel.26

Figure 3shows a typical example of the magnetoresist-ance RxxðBÞ and Hall resistance (Rxy) measured at 1.8 K.

SdH oscillations and Hall plateau are clearly visible over the magnetic field range betweenB¼ 2 and 7 T and no higher harmonics are apparent in the SdH oscillations. The overall positiveRxyðBÞ indicates that the contribution is mainly from

electrons. At a high magnetic field,RxyðBÞ exhibits plateau

andRxxðBÞ is vanishing, which is the hallmark of the

quan-tum hall effect (QHE).7At least two well-defined plateaus with values (h=2e2_{¼ 12893 X) and (h=6e}2_{¼ 4300 X),}

followed by a developing (h=10e2_{¼ 2581X) plateau, are}

observed before the QHE features transform into SdH oscil-lations at a lower magnetic field. The plateaux numbers observed follow the 2þ 4v (v ¼ 0, 1, 2,. is the filling factor of the quantum Hall states) pattern which confirms that our sample is monolayer graphene.18,27,28

In order to check the 2D nature of the electron gas giv-ing rise to the quantum oscillations in magnetoresistance, measurements were also performed as a function of the angle h between the normal to the plane of the 2D electron gas and the applied magnetic field. It was found that the peak posi-tion shift with a factor ofcosh and the oscillations disappear at h¼ 808_(Fig.₄_{). This observation is a characteristic of 2D}

electron gas.29

Figures 5(a) and 5(b) show typical examples for the magnetoresistance RxxðBÞ measurements in the graphene

sample for different lattice temperatures at a fixed electric

F¼ 4.43 V m1, and field and for different electric fields at a fixed lattice temperature TL¼ 1.8 K, respectively. The data

clearly show the decrease in the amplitude of the SdH oscil-lations with an increasing lattice temperature or electric field. The peak positions of the oscillations are also shifted to a higher magnetic field with an increasing temperature or elec-tric field. Similar behaviors have also been observed for monolayer graphene in the literature.7,18 The magnetic field of the initial value of the Hall plateau and the zero value of RxxðBÞ at a high magnetic field shift to a higher magnetic

field with an increasing temperature and electric field. The period of the SdH oscillations has been obtained from the Fourier analysis. The Fourier analysis of the SdH oscillations (Fig. 6) confirms that only the first subband is populated and that the contribution of higher harmonics is in-significant. The 2D carrier density (N2D) of the graphene

sample can be calculated using the oscillation period30,31

D 1 Bn ¼ e phN2D ; (4)

wheree and hð¼ h=2pÞ are the electron charge and Planck’s constant, respectively. The results for the graphene sample is

FIG. 2. Temperature dependence of the sheet carrier density (NH) and Hall

mobility (l_H) of electrons in the graphene.

FIG. 3. The magnetoresistance (RxxðBÞ) and Hall resistance (RxyðBÞ) for

gra-phene sample measured at 1.8 K. The horizontal lines correspond toh=ve2

values. The QHE in the electron gas is demonstrated by only one quantized plateau inRxywith vanishingRxxin the corresponding magnetic field regime.

FIG. 4. The normalized magnetoresistanceRxxðBÞ=RxxðB ¼ 0Þ measured at

1.8 K as a function of the angle h between the normal to the plane of the 2D electron gas and the applied magnetic field. The vertical arrows and dotted lines indicate the angle dependence of the peak position and shifting of the h¼ 0 peak position of the cosh function, respectively.

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N2D¼ 1:51 1015m2. The 2D carrier density is found to

be essentially independent of the lattice temperature in the range from TL¼ 1.8 to 250 K. In our sample, the ratio

NH=N2Dis about one within the experimental accuracy. This

result indicates that only 2D electron densities contribute to the transport properties and, therefore, parallel conduction is negligible. The effective massm can be extracted from the temperature dependence of the SdH amplitude at a constant magnetic field using30,31

AðT; BnÞ AðT0; BnÞ ¼Tsinhð2p 2_k BT0m=heBnÞ T0sinhð2p2kBTm=heBnÞ ; (5)

whereAðT; BnÞ and AðT0; BnÞ are the amplitudes of the

oscil-lation peaks observed at a magnetic field Bnand at

tempera-turesT and T0. The relative amplitudes, AðT; BnÞ=AðT0; BnÞ,

are plotted in Fig.7(a). The relative amplitude decreases with an increasing temperature as a result of thermal damping.5,32 The effective mass of 2D electrons is then determined by fit-ting the experimental data for the temperature dependence of AðT; BnÞ=AðT0; BnÞ to Eq.(5). We findm¼ 0:012m0 (m0is

the free electron mass). We obtained similar values for all the magnetic fields of measurements, thereby strongly suggesting that the effective mass is essentially independent of the mag-netic field. Our electron effective mass for the monolayer gra-phene sample agrees with the calculated effective mass m¼ ðh=vFÞ ffiffiffiffiffiffiffiffiffiffiffipN2D

p

, whereN2D is obtained from the

meas-ured SdH period and vF¼ 1.1 106ms1is the band velocity

adopted from the literature.7,8

The SdH oscillations and classical low-field Hall-effect measurements allow for the determination of both the quan-tum and transport lifetimes of the electrons in the graphene and hence to investigate the relative importance of various scattering mechanisms. The quantum lifetime can be deter-mined from the magnetic-field dependence of the amplitude of the SdH oscillations (i.e., Dingle plots) at a constant temperature provided that the electron effective mass is known26,30,31 ln AðT; BnÞ:B 1=2 n :sinhðvÞ v ¼ C pm esq 1 Bn ; (6)

FIG. 5. The effects of temperature (a) and applied electric fields (b) on the SdH oscillations for the graphene sample.

FIG. 6. The fast Fourier spectrum of the oscillations. There is no evidence for the population of higher subbands or for any contribution from higher harmonics.

FIG. 7. (a) Temperature and (b) electric-field dependencies of the normal-ized amplitude of the oscillation peak atBnmeasured in the graphene

sam-ple. The data points represented by the full circles correspond to the SdH oscillations. The full curve in (a) is the best fit of Eq.(5)to the experimental data. The full curve in (b) is intended as a guide.

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whereC is a constant. Figure8shows typical examples of the Dingle plots for the samples investigated. There is good agreement between the experimental data and the straight line described by Eq. (6). The quantum lifetime obtained from the slope of the Dingle plot is sq¼ 23 fs. These values

remain constant within 2% in the whole temperature and magnetic-field ranges of the measurements. The ratio of the quantum to transport lifetime, st=sq, in our samples is bigger

than unity (st=sq¼ 1:36). Theoretical calculations relating

the 2D single-particle scattering time (quantum lifetime) to the momentum relaxation time (transport lifetime) predict the st=sq ratio of equal to or less than unity for wide-angle

scattering and greater than unity for small-angle scattering in the extreme quantum limit for single subband occupancy.26 This implies that in our sample the electron scattering with small-angle scattering, such as long-range Coulomb disorder scattering, is on average forward displaced in momentum space. A similar result is also attributed to a graphene structure.33

Assuming that the change in the SdH amplitude with an applied electric field can be described in terms of electric-field induced electron heating, the temperatureT in Eq. (5) can be replaced by the electron temperatureTe.2,3,5,26

There-fore,Tecan be determined by comparing the relative

ampli-tudes of the SdH oscillations measured as functions of the lattice temperature (T¼ TL) and the applied electric field (F)

using2,3,5 AðTL; BnÞ AðTL0; BnÞ F¼F0 ¼ AðF; BnÞ AðF0; BnÞ TL¼TL0 : (7)

HereAðF; BnÞ and AðF0; BnÞ are the amplitudes of the

oscilla-tion peaks observed at a magnetic field Bn and at electric

fieldsF and F0, respectively. In order to obtain the electron

temperature from the lattice temperature and electric-field dependencies of the amplitude of the SdH oscillations, the quantum lifetime has to be independent of both the lattice temperature and the applied electric field. Figure7(b)shows the amplitudes of the SdH oscillations, normalized as described by Eq. (6), as functions of F for the monolayer graphene sample. In Fig.7, only the relative amplitudes at a given magnetic fieldBnare shown for clarity. A similar

anal-ysis that was conducted for all the SdH peaks that were

observed in the magnetic field range from 2 to 11 T has estab-lished that the relative amplitudes of SdH oscillations (and hence the electron temperatures) in our samples are essen-tially independent of a magnetic field. This indicates that the magnetic field used in our experiments does not significantly alter the energy relaxation processes of hot electrons.

Electron temperatures (Te) for the monolayer graphene

sample as obtained by directly comparing the curves similar to those in Figs.7(a)and7(b)are plotted as a function of the applied electric field in Fig. 9. The SdH oscillations meas-ured for the monolayer graphene sample decrease rapidly with the increase of the applied electric field and become vanishingly small for F > 35 kV m1 [see Figs. 5(b) and 7(b)]. The electron temperature determined for this sample rises quickly with increasingF.

In the steady state, the power loss from hot electrons by the emission of acoustic phonons is equal to the power sup-plied by the apsup-plied electric field, which can be calculated using the energy balance equation2,3

P¼ elHF

2_; ₍₈₎

whereP and F are the energy loss (or energy supply) rate per electron and applied electric field, respectively. In order to obtain the electron temperature from the lattice temperature and electric-field dependencies of the amplitude of the SdH oscillations, both the quantum lifetime and the transport mo-bility have to be independent of both the lattice temperature and the applied electric field. It is found that l_His independ-ent of both the lattice temperature and the applied electric field in the ranges of the measurements. The power loss ver-sus electron temperature is plotted in Fig. 10. The electron temperature dependence of the power loss was found to be similar with the previous reports on energy relaxation by hot electrons in graphene samples.9,18

The variation of the power loss per electron with elec-tron temperature in the acoustic phonon regime has often been approximated by the relationship

P¼ AðTc e T

c

L0Þ; (9)

where A is the proportionality constant, which depends on the elastic moduli of the matrix, the coupling constants, and the 2D carrier density. Theoretical calculations of the

FIG. 8. Determination of the quantum lifetime in the graphene sample. The data points represented by symbols and the straight lines are the least-squares fits of Eq.(6)to the experimental data, respectively.

FIG. 9. Electron temperature (Te) versus the applied electric field (F) for the

graphene sample.

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acoustic phonon assisted energy loss rates of hot electrons in a 2D electron gas of single-subband occupancy predict c¼ 1 at high temperatures (when Maxwell-Boltzmann statis-tics is applicable and equipartition is assumed) and c¼ 3 (unscreened piezoelectric scattering), c¼ 5 (unscreened deformation potential and heavily-screened piezoelectric scatterings), and c¼ 7 (heavily screened deformation poten-tial scattering) at low temperatures (see, for instance, Ref.4). We found the exponent 3 in the range 1:8 <Te< 50 K and

5 in the range 50 <Te< 130 K by fitting Eq. (9) to the

experimental data (Fig. 10). In all cases, a constant value for the exponent c is obtained over the temperature range (Fig.10). This indicates that the experiments were carried out in the low-temperature regime and that the energy relaxation is due to acoustic phonon emission via mixed unscreened pie-zoelectric and deformation potential interactions.

The energy relaxation time (sE) for intrasubband

proc-esses can be obtained from the power loss measurements using4 P¼hhxi sE ðkBTe kBTLÞ kBTe ; (10) wherehhxi ¼ 21=2_hV

SkFis the acoustic-phonon energy

aver-aged over the Fermi surface. Figure 10 shows the energy relaxation time as a function of the electron temperature

for the graphene sample studied. The electron temperature dependence of the energy relaxation time was found to be similar with the previous reports on energy relaxation by hot electrons in graphene samples.9,18 However, as can be seen in Fig. 11, the energy relaxation due to acoustic phonons becomes faster at higher electron temperatures.

IV. CONCLUSIONS

The carrier density (N2D) and effective mass (m) for

electrons in the monolayer graphene have been determined from the Shubnikov de Haas (SdH) oscillations. The two-dimensional (2D) carrier density and the m have been obtained from the periods of the SdH oscillations and the tem-perature dependencies of the SdH amplitude, respectively.

The energy-loss rates, in the acoustic phonon regime, of 2D electrons in a monolayer graphene have also been investi-gated using SdH effect measurements. The experimental results were compared with the predictions of the current the-oretical models for power loss in semiconductors. The energy relaxation of electrons is due to acoustic phonon emission via piezoelectric and deformation-potential scattering.

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Effective mass of electron in monolayer graphene : Electron-phonon interaction

Effective mass of electron in monolayer

graphene: Electron-phonon interaction

E Tiras, S Ardali, T Tiras, E Arslan, S Cakmakyapan, O Kazar, Jawad Hassan, Erik Janzén

and E Ozbay

Linköping University Post Print

N.B.: When citing this work, cite the original article.

Original Publication:

E Tiras, S Ardali, T Tiras, E Arslan, S Cakmakyapan, O Kazar, Jawad Hassan, Erik Janzén

and E Ozbay, Effective mass of electron in monolayer graphene: Electron-phonon interaction,

2013, Journal of Applied Physics, (113), 4.

http://dx.doi.org/10.1063/1.4789385

Copyright: American Institute of Physics (AIP)

http://www.aip.org/

Postprint available at: Linköping University Electronic Press

Effective mass of electron in monolayer graphene: Electron-phonon

interaction

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Effective mass of electron in monolayer graphene: Electron-phonon

interaction